Mark,

There is an explanation for this term that is essentially what Martina wrote in my book:

McCulloh, I., Armstrong, H., & Johnson, A.N. (2013) Social Network Analysis With Applications. Hoboken, NJ: Wiley.

In Chapter 4, I lay out six social forces that contribute to tie formation and the references that support them. Each of these forces have supporting terms in an ergm that allow you to test for the presence and significance of these social forces in a particular network.

I don't think the explanation in my book will add anything to what Martina explained, but if you are looking for a reference, it is in there.

Regards,

Ian

Ian McCulloh

On Mon, Nov 10, 2014 at 2:49 PM, Lubell, Mark <[log in to unmask]> wrote:

***** To join INSNA, visit http://www.insna.org *****

Dear Martina, all:

Thank you that intuitive interpretation. I would indeed like to see the spreadsheet, if it is easy for you to make it available. And some further off-list dialog would be great, to make sure we get it absolutely correct.

But I will reply here to the broader list, because I've been looking at papers that are interpreting the GWDEGREE parameter and the interpretations are really all over the place. The applied literature appears to be a big mess and the different interpretations are leading to different conclusions.

I would love if somebody wrote a paper with simulations etc entitled "What is Geometrically Weighted Degree Distribution, Really?"

Mark

________________________________________

From: martina morris [[log in to unmask]]

Sent: Monday, November 10, 2014 11:38 AM

To: Lubell, Mark

Cc: Social Networks Discussion Forum

Subject: Re: Interpretation of GWDEGREE

Hi Mark,

These are complicated terms to understand, and we do need to make a

more intuitive explanation available to folks. Here's how I like to think

about this.

There are two parameters in the gw-terms, and the overall effect on the

odds of a tie is a product of the two, given the way the statistic is

constructed. That's why these are called "curved terms".

You can think of the gwdegree term as having the form

beta * f(y, alpha)

This has the usual form parameter*statistic, except that there is a second

parameter, alpha, in the statistic itself. The statistic essentially sums

the number of nodes of each degree, except that alpha modifies the value

of that number, as a function of degree.

Alpha essentially imposes a rate of decay by degree, so the higher degree

nodes contribute less to the statistic than the lower. It can be

interpreted as the declining marginal return for each additional tie (or

additional shared partner for gw(e/n/d)sp). So yes, this does relate to

the preferential attachment concept (more below).

Beta controls the overall propensity for degree (or shared partners).

A good way to start to interpret the parameters is to set alpha=0, and

look at the change statistics (you can do this by calculating the f(y,

alpha) statistic with and without a proposed tie). Setting alpha=0 has

the effect of making only the first tie for a node count as a change; so

the possible values of the change statistic are

0 (if both nodes already have other ties),

1 (if one node was an isolate), and

2 (if both nodes were isolates).

Beta then multiplies this, so it can be interpreted as how the odds of a

tie change, as a function of the change in the number of nodes that are no

longer isolates when it is toggled on.

Of course, interpretation depends on the other terms in the model, and in

general you would have an edges term in to control overall density. In

that case, beta would reflect a propensity against/for isolates (for

positive/negative estimates respectively), relative to a random graph

with this density.

When alpha > 0, there is no discontinuity at 1 vs more, but instead a

continuous decline in the value of additional partners, where the rate of

decline falls as alpha increases. For alpha=inf, there is no declining

marginal return, the odds of a tie don't depend on the degrees of the

nodes (and for shared partners, you're back to the triangle term).

So, in answer to your question, it's the alpha parameter that is the

"anti-preferential attachment" component. As it varies from 0 to inf., it

never represents preferential attachment -- at inf., ties are just

independent of degree. But the smaller the value of alpha, the more

anti-preferential the degree distribution will be.

I found it helped me to understand these terms by making up an excel

spreadsheet to calculate the term itself, and the change statistics. If

you think something like this might help, I can clean mine up and make it

available.

best,

Martina

On Mon, 10 Nov 2014, Lubell, Mark wrote:

> ***** To join INSNA, visit http://www.insna.org *****

>

> Dear SOCNET:

>

> My research group is having an internal debate about how to interpret the geometrically weighted degree parameter for ERGM models, as implemented in Statnet. If anybody has a good paper or presentation discussing interpretation (beyond the various papers introducing the calculation and estimation), I would love to know about them.

>

> In particular, is GWDEGREE an anti-preferential attachment term such that a positive coefficient produces a low variance degree distribution, or does a positive coefficient produce a high variance degree distribution with a centralized network? And if you have a low variance degree distribution....what is the best way to think about the social processes generating a decentralized network?

>

> Thanks, Mark Lubell

> UC Davis

>

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